url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_left_cancel | [58, 1] | [59, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = a + c
⊢ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.add_right_cancel | [61, 1] | [62, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b c : R
h : a + b = c + b
⊢ a = c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | have h : a * 0 + a * 0 = a * 0 + 0 := by
rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 = 0 | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | rw [add_left_cancel h] | R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
h : a * 0 + a * 0 = a * 0 + 0
⊢ a * 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.mul_zero | [64, 1] | [67, 25] | rw [← mul_add, add_zero, add_zero] | R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 + a * 0 = a * 0 + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a * 0 + a * 0 = a * 0 + 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.zero_mul | [69, 1] | [70, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 0 * a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_eq_of_add_eq_zero | [72, 1] | [73, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ -a = b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.eq_neg_of_add_eq_zero | [75, 1] | [76, 8] | sorry | R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a b : R
h : a + b = 0
⊢ a = -b
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [78, 1] | [80, 16] | apply neg_eq_of_add_eq_zero | R : Type u_1
inst✝ : Ring R
⊢ -0 = 0 | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ -0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_zero | [78, 1] | [80, 16] | rw [add_zero] | case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
R : Type u_1
inst✝ : Ring R
⊢ 0 + 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.neg_neg | [82, 1] | [83, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ - -a = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.self_sub | [105, 1] | [106, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ a - a = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.one_add_one_eq_two | [108, 1] | [109, 11] | norm_num | R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
⊢ 1 + 1 = 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyRing.two_mul | [111, 1] | [112, 8] | sorry | R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : Ring R
a : R
⊢ 2 * a = a + a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_right_inv | [134, 1] | [135, 8] | sorry | G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_one | [137, 1] | [138, 8] | sorry | G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S2_Basics/S02_Proving_Identities_in_Algebraic_Structures.lean | MyGroup.mul_inv_rev | [140, 1] | [141, 8] | sorry | G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
inst✝ : Group G
a b : G
⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [61, 1] | [62, 8] | sorry | x✝ y x : ℝ
⊢ x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [64, 1] | [65, 8] | sorry | x✝ y x : ℝ
⊢ -x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_add | [67, 1] | [68, 8] | sorry | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [70, 1] | [71, 8] | sorry | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [73, 1] | [74, 8] | sorry | x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ |x| < y ↔ -y < x ∧ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S5_Topology/S03_Topological_Spaces.lean | aux | [104, 1] | [108, 8] | sorry | X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X✝ : Type u_1
Y✝ : Type u_2
X : Type u_3
Y : Type u_4
A : Type u_5
inst✝ : TopologicalSpace X
c : A → X
f : A → Y
x : X
F : Filter Y
h : Tendsto f (comap c (𝓝 x)) F
V' : Set Y
V'_in : V' ∈ F
⊢ ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V'
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | inverse_spec | [158, 1] | [160, 32] | rw [inverse, dif_pos h] | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (inverse f y) = y | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (inverse f y) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | inverse_spec | [158, 1] | [160, 32] | exact Classical.choose_spec h | α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
inst✝ : Inhabited α
P : α → Prop
h✝ : ∃ x, P x
f : α → β
y : β
h : ∃ x, f x = y
⊢ f (choose h) = y
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | intro f surjf | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
⊢ ∀ (f : α → Set α), ¬Surjective f
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | let S := { i | i ∉ f i } | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | rcases surjf S with ⟨j, h⟩ | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | have h₁ : j ∉ f j := by
intro h'
have : j ∉ f j := by rwa [h] at h'
contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | have h₂ : j ∈ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf :... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | sorry | case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf :... | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
⊢ j ∈ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | have h₃ : j ∉ S | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Se... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | sorry | case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Se... | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h₃
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
⊢ j ∉ S
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inh... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | contradiction | case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h₁ : j ∉ f j
h₂ : j ∈ S
h₃ : j ∉ S
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | intro h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
⊢ j ∉ f j
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | have : j ∉ f j := by rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | contradiction | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
this : j ∉ f j
⊢ False
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S4_Sets_and_Functions/S02_Functions.lean | Cantor | [178, 1] | [190, 16] | rwa [h] at h' | α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
β : Type u_2
inst✝ : Inhabited α✝
P : α✝ → Prop
h✝ : ∃ x, P x
α : Type u_3
f : α → Set α
surjf : Surjective f
S : Set α := {i | i ∉ f i}
j : α
h : f j = S
h' : j ∈ f j
⊢ j ∉ f j
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | let y := f c | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hy : Tendsto f (𝓝 c) (𝓝 y) := by
rw [← (@nhdsWithin_eq_nhds _ _ c (closedBall c R)).mpr (closedBall_mem_nhds _ h0)]
exact ContinuousOn.continuousWithinAt hf <| mem_closedBall_self (le_of_lt h0) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [← sub_eq_zero, ← norm_le_zero_iff] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ (∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • f c
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | refine' le_of_forall_le_of_dense fun ε ε0 => _ | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ 0 | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ ε | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ 0
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | obtain ⟨α, α0, hα⟩ := (nhds_basis_ball.tendsto_iff (nhds_basis_ball)).1 hy _
(div_pos ε0 Real.two_pi_pos) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ ε | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f c‖ ≤ ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | set δ := α/2 | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • f ... | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
⊢... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have δ0 := half_pos α0 | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2... | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hδ := fun z hz => hα z ((@closedBall_subset_ball ℂ _ c (α/2) α (by linarith)) hz) | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
⊢ ‖(∮ (z : ℂ) in C(c, R), (z - c... | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2),... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | obtain ⟨r, hr0, hrδ, hrR⟩ : ∃ r, 0 < r ∧ r ≤ δ ∧ r ≤ R :=
⟨min δ R, lt_min δ0 h0, min_le_left _ _, min_le_right _ _⟩ | case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2),... | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hsub : closedBall c R \ ball c r ⊆ closedBall c R := diff_subset (closedBall c R) (ball c r) | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hsub' : ball c R \ closedBall c r ⊆ ball c R \ {c} :=
diff_subset_diff_right (singleton_subset_iff.2 <| mem_closedBall_self hr0.le) | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hzne : ∀ z ∈ sphere c r, z ≠ c := fun z hz =>
ne_of_mem_of_not_mem hz fun h => hr0.ne' <| dist_self c ▸ Eq.symm h | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ clo... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [← (@nhdsWithin_eq_nhds _ _ c (closedBall c R)).mpr (closedBall_mem_nhds _ h0)] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ Tendsto f (𝓝 c) (𝓝 y) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ Tendsto f (𝓝[closedBall c R] c) (𝓝 y) | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ Tendsto f (𝓝 c) (𝓝 y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | exact ContinuousOn.continuousWithinAt hf <| mem_closedBall_self (le_of_lt h0) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ Tendsto f (𝓝[closedBall c R] c) (𝓝 y) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
⊢ Tendsto f (𝓝[closedBall c R] c) (𝓝 y)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | linarith | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
z : ℂ
hz : z ∈ closedBall c (α / 2)
⊢ α / 2 < α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | congr 2 | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | refine circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0
hrR countable_empty (hf.mono hsub) fun z hz => hdiff z ?_ | case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z... | case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | apply mem_of_subset_of_mem _ hz | case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [diff_empty] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | exact diff_subset (ball c R) (closedBall c r) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | simp only [circleIntegral.integral_smul_const, ne_eq, hr0.ne', not_false_eq_true,
circleIntegral.integral_sub_center_inv] | case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.e_a
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | simp only [smul_sub] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | have hc' : ContinuousOn (fun z => (z - c)⁻¹) (sphere c r) :=
(continuousOn_id.sub continuousOn_const).inv₀ fun z hz => sub_ne_zero.2 <| hzne _ hz | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [circleIntegral.integral_sub] <;> refine' (hc'.smul _).circleIntegrable hr0.le | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | case hf
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ba... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | exact hf.mono <| (sphere_subset_closedBall).trans (closedBall_subset_closedBall hrR) | case hf
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ba... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | exact continuousOn_const | case hg
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ba... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | refine' circleIntegral.norm_integral_le_of_norm_le_const hr0.le fun z hz => _ | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | specialize hzne z hz | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [mem_sphere, dist_eq_norm] at hz | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rw [norm_smul, norm_inv, hz, ← dist_eq_norm] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | refine' mul_le_mul_of_nonneg_left (le_of_lt <| hδ _ _) (inv_nonneg.2 hr0.le) | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | rwa [mem_closedBall_iff_norm, hz] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | field_simp [hr0.ne', Real.two_pi_pos.ne'] | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Cauchy_formula | [14, 1] | [73, 13] | ac_rfl | c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 : 0 < α / 2
hδ : ∀ z ∈ closedBall c (α / 2), f z ∈ ball y (ε ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : ℂ
R : ℝ
h0 : 0 < R
f : ℂ → ℂ
hf : ContinuousOn f (closedBall c R)
hdiff : ∀ z ∈ ball c R, DifferentiableAt ℂ f z
y : ℂ := f c
hy : Tendsto f (𝓝 c) (𝓝 y)
ε : ℝ
ε0 : 0 < ε
α : ℝ
α0 : 0 < α
hα : ∀ x ∈ ball c α, f x ∈ ball y (ε / (2 * π))
δ : ℝ := α / 2
δ0 ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | Colloquia/ConversationWithLean/Colloquium.lean | Odd_Odd_Even | [80, 1] | [80, 84] | sorry | n m : ℕ
hn : Odd n
hm : Odd m
⊢ Even (n + m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n m : ℕ
hn : Odd n
hm : Odd m
⊢ Even (n + m)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | intro ε εpos | a : ℝ
⊢ ConvergesTo (fun x => a) a | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℝ
⊢ ConvergesTo (fun x => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | use 0 | a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
a ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | intro n nge | case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
⊢ ∀ n ≥ 0, |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | rw [sub_self, abs_zero] | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ |(fun x => a) n - a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_const | [9, 1] | [14, 13] | apply εpos | case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a ε : ℝ
εpos : ε > 0
n : ℕ
nge : n ≥ 0
⊢ 0 < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | intro ε εpos | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b) | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
⊢ ConvergesTo (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | dsimp | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => s n + t n) n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ε2pos : 0 < ε / 2 := by linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | rcases cs (ε / 2) ε2pos with ⟨Ns, hs⟩ | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | rcases ct (ε / 2) ε2pos with ⟨Nt, ht⟩ | case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | use max Ns Nt | case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∃ N, ∀ n ≥ N, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | intro n hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
⊢ ∀ n ≥ max Ns Nt, |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ngeNs : n ≥ Ns := le_of_max_le_left hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
⊢ |s n + t n - (a + b)| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | have ngeNt : n ≥ Nt := le_of_max_le_right hn | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
⊢ |s n + t n - (a + b)| < ε
TACTI... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | calc
|s n + t n - (a + b)| = |s n - a + (t n - b)| := by
congr
ring
_ ≤ |s n - a| + |t n - b| := (abs_add _ _)
_ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt))
_ = ε := by norm_num | case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| < ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a ... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | linarith | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
⊢ 0 < ε / 2
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | congr | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| = |s n - a + (t n - b)| | case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a + b) = s n - a + (t n - b) | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ |s n + t n - (a + b)| =... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | ring | case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a + b) = s n - a + (t n - b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ s n + t n - (a... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_add | [16, 1] | [34, 25] | norm_num | s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ ε / 2 + ε / 2 = ε | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : ℕ → ℝ
a b : ℝ
cs : ConvergesTo s a
ct : ConvergesTo t b
ε : ℝ
εpos : ε > 0
ε2pos : 0 < ε / 2
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / 2
Nt : ℕ
ht : ∀ n ≥ Nt, |t n - b| < ε / 2
n : ℕ
hn : n ≥ max Ns Nt
ngeNs : n ≥ Ns
ngeNt : n ≥ Nt
⊢ ε / 2 + ε / 2 = ε
TACTI... |
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | by_cases h : c = 0 | s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a) | case pos
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | have acpos : 0 < |c| := abs_pos.mpr h | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a) | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | intro ε εpos | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a) | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
⊢ ConvergesTo (fun n => c * s n) (c * a)
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | dsimp | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |(fun n => c * s n) n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | rcases cs (ε / |c|) εcpos with ⟨Ns, hs⟩ | case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | use Ns | case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∃ N, ∀ n ≥ N, |c * s n - c * a| < ε
TACTIC:
|
https://github.com/fpvandoorn/LeanInRome.git | 55e064179515cdc8f96fd8e82b2c106fb80e5c0e | LeanInRome/S3_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean | C03S06.convergesTo_mul_const | [36, 1] | [54, 53] | intro n ngt | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε | case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
n : ℕ
ngt : n ≥ Ns
⊢ |c * s n - c * a| < ε | Please generate a tactic in lean4 to solve the state.
STATE:
case h
s : ℕ → ℝ
a c : ℝ
cs : ConvergesTo s a
h : ¬c = 0
acpos : 0 < |c|
ε : ℝ
εpos : ε > 0
εcpos : 0 < ε / |c|
Ns : ℕ
hs : ∀ n ≥ Ns, |s n - a| < ε / |c|
⊢ ∀ n ≥ Ns, |c * s n - c * a| < ε
TACTIC:
|
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